# Timber Rafter roof design: Complete guide of structural calculations

Last updated: November 8th, 2022

Another day, another design article about timber roofs.🤗

After we have covered the structural designs of the collar beam roof and the purlin roof – it is now time to look at the timber rafter roof and design it.🕵️

In this article we’ll show exactly, step-by step, how to dimension and design the elements of the rafter roof according to the Timber Eurocode EN 1995-1-1:2004.

Not much more talk, let’s dive into it.

## 🙋♀️ What is a rafter roof?

The rafter roof consists of 2 rafters which are inclined and connected to each other at the top with a ridge board. The rafters are statically speaking beams and have usually a connection/support at the bottom (floor) and the top (2nd rafter or ridge board). Depending on the design the rafters can also be cantilievered to create an overhang of the roof.

As already mentioned, there is different ways to build the rafter roof, meaning that the different elements can be built with different materials and systems.

One example of the rafter roof type can be seen in the next picture where a timber beam is chosen as the ceiling joist which takes up the horizontal support forces of the rafters in, mostly, tension.

A steel wind bracing strap is furthermore chosen as the wind bracing system.

We haven’t covered wind bracing systems yet, – how they work, why we need them – but would you be interested in learning more? Let me know in the comments below.

## 👆 Static system of the rafter roof

The static system of the rafter roof is built up by 2 inclined **rafters modelled as beams** and connected to each other at the top with a hinge.

Those beams are supported with **pinned supports** at its lowest point or – in case of a **cantilevered overhang** of the roof – close to the lowest point.

The static system of the rafter roof is visualized in the next picture.

To not lose context – the 2D static system represents the following rafters.

But it can also represent any other section of rafters and beam. The spacing between the rafters is set to 1 m.

The rafter roof can of course also have different layouts with wider spans or steeper inclination.

## ⬇️ Characteristic Loads – Rafter roof

The loads will not be derived in this article. We explained the calculation of dead, live, wind and snow loads for pitched roofs thoroughly in previous articles.

The defined load values are estimations from the previous calculations.

$g_{k}$ | 1.08 kN/m2 | Characteristic value of dead load |

$q_{k}$ | 1.0 kN/m2 | Characteristic value of live load |

$s_{k}$ | 0.53 kN/m2 | Characteristic value of snow load |

As we also discussed in the article about the characteristic snow load, there are 3 different load cases, where only half of the value is applied on one pitched side but the full value on the other.

However, due to simplicity we only consider load case 1 in this tutorial which applies $s_{k} = 0.53 kN/m^2$ on both rafters.

We will split up the wind load from the above table due to the complexity of the wind with its wind areas and directions.

In this calculation, we will only focus on the external wind pressure for areas of 10 m2.

**Wind direction front**

$w_{k.F}$ | -0.25(/0.35) kN/m2 | Characteristic value of wind load Area F |

$w_{k.G}$ | -0.25(/0.35) kN/m2 | Characteristic value of wind load Area G |

$w_{k.H}$ | -0.1(/0.2) kN/m2 | Characteristic value of wind load Area H |

$w_{k.I}$ | -0.2(/0.0) kN/m2 | Characteristic value of wind load Area I |

$w_{k.J}$ | -0.25(/0.0) kN/m2 | Characteristic value of wind load Area J |

**Wind direction side**

$w_{k.F}$ | -0.55 kN/m2 | Characteristic value of wind load Area F |

$w_{k.G}$ | -0.7 kN/m2 | Characteristic value of wind load Area G |

$w_{k.H}$ | -0.4 kN/m2 | Characteristic value of wind load Area H |

$w_{k.I}$ | -0.25 kN/m2 | Characteristic value of wind load Area I |

The following picture presents the static system of the collar beam roof with its line loads applied.

The section that is presented in *Figure: Rafter roof | 2D static system representing rafters and beams *is used for this example.

Due to simplicity, this tutorial looks only at the wind load from the side. Therefore, the **wind load** **$w_{k.I} = -0.25 kN/m^2$** is applied to both rafters.

## ➕ Load combinations – Rafter roof

Luckily we have already written an extensive article about what load combinations are and how we use them. In case you need to brush up on it you can read the blog post here.

We choose to include $w_{k.I.}$ = -0.25 kN/m2 as the wind load in the load combinations, as this is the wind load that is applied to the section we look at, and to keep the calculation clean.

In principle, you should consider all load cases. However, with a bit more experience, you might be able to exclude some of the values.

In modern FE programs, multiple values for the wind load can be applied and load combinations automatically generated. So the computer is helping us a lot.

Just keep in mind that you should include all wind loads but because of simplicity we do only consider 1 value in this article😁.

**ULS Load combinations**

I know you might not understand what that means when you do load combinations the first time, but we did a whole article about what loads exist and how to apply them on a pitched roof 😎

LC1 | $1.35 * 1.08 \frac{kN}{m^2} $ |

LC2 | $1.35 * 1.08 \frac{kN}{m^2} + 1.5 * 1.0 \frac{kN}{m^2}$ |

LC3 | $1.35 * 1.08 \frac{kN}{m^2} + 1.5 * 1.0 \frac{kN}{m^2} + 0.7 * 1.5 * 0.53 \frac{kN}{m^2}$ |

LC4 | $1.35 * 1.08 \frac{kN}{m^2} + 0 * 1.5 * 1.0 \frac{kN}{m^2} + 1.5 * 0.53 \frac{kN}{m^2}$ |

LC5 | $1.35 * 1.08 \frac{kN}{m^2} + 1.5 * 1.0 \frac{kN}{m^2} + 0.7 * 1.5 * 0.53 \frac{kN}{m^2} + 0.6 * 1.5 * (-0.25 \frac{kN}{m^2}) $ |

LC6 | $1.35 * 1.08 \frac{kN}{m^2} + 0 * 1.5 * 1.0 \frac{kN}{m^2} + 1.5 * 0.53 \frac{kN}{m^2} + 0.6 * 1.5 * (-0.25 \frac{kN}{m^2}) $ |

LC7 | $1.35 * 1.08 \frac{kN}{m^2} + 0 * 1.5 * 1.0 \frac{kN}{m^2} + 0.7 * 1.5 * 0.53 \frac{kN}{m^2} + 1.5 * (-0.25 \frac{kN}{m^2}) $ |

LC8 | $1.35 * 1.08 \frac{kN}{m^2} + 1.5 * 0.53 \frac{kN}{m^2} $ |

LC9 | $1.35 * 1.08 \frac{kN}{m^2} + 1.5 * (-0.25 \frac{kN}{m^2}) $ |

LC10 | $1.35 * 1.08 \frac{kN}{m^2} + 1.5 * 1.0 \frac{kN}{m^2} + 0.6 * 1.5 * (-0.25 \frac{kN}{m^2}) $ |

LC11 | $1.35 * 1.08 \frac{kN}{m^2} + 1.5 * (-0.25 \frac{kN}{m^2}) + 0.7 * 1.5 * 0.53 \frac{kN}{m^2} $ |

LC12 | $1.35 * 1.08 \frac{kN}{m^2} + 1.5 * 0.53 \frac{kN}{m^2} + 0.6 * 1.5 * (-0.25 \frac{kN}{m^2})$ |

**Characteristic SLS Load combinations**

LC1 | $1.08 \frac{kN}{m^2} $ |

LC2 | $1.08 \frac{kN}{m^2} + 1.0 \frac{kN}{m^2}$ |

LC3 | $1.08 \frac{kN}{m^2} + 1.0 \frac{kN}{m^2} + 0.7 * 0.53 \frac{kN}{m^2}$ |

LC4 | $1.08 \frac{kN}{m^2} + 1.0 \frac{kN}{m^2} + 0.6 * (-0.25 \frac{kN}{m^2})$ |

LC5 | $1.08 \frac{kN}{m^2} + 1.0 \frac{kN}{m^2} + 0.7 * 0.53 \frac{kN}{m^2} + 0.6 * (-0.25 \frac{kN}{m^2}) $ |

LC6 | $1.08 \frac{kN}{m^2} + 0 * 1.0 \frac{kN}{m^2} + 0.53 \frac{kN}{m^2} + 0.6 * (-0.25 \frac{kN}{m^2}) $ |

LC7 | $1.08 \frac{kN}{m^2} + 0 * 1.0 \frac{kN}{m^2} + 0.7 * 0.53 \frac{kN}{m^2} + (-0.25 \frac{kN}{m^2}) $ |

LC8 | $1.08 \frac{kN}{m^2} + 0.53 \frac{kN}{m^2}$ |

LC9 | $1.08 \frac{kN}{m^2} + (-0.25 \frac{kN}{m^2}) $ |

LC10 | $1.08 \frac{kN}{m^2} + 1.0 \frac{kN}{m^2} + 0.6 * (-0.25 \frac{kN}{m^2}) $ |

LC11 | $1.08 \frac{kN}{m^2} + (-0.25 \frac{kN}{m^2}) + 0.7 * 0.53 \frac{kN}{m^2} $ |

LC12 | $1.08 \frac{kN}{m^2} + 0 * 1.0 \frac{kN}{m^2} + 0.53 \frac{kN}{m^2}$ |

LC13 | $1.08 \frac{kN}{m^2} + 0 * 1.0 \frac{kN}{m^2} + (-0.25 \frac{kN}{m^2})$ |

## 🪵 Rafter timber material

For this blog post/tutorial we are choosing a Structural timber C24. More comments on which timber material to pick and where to get the properties from were made here.

The following characteristic strength and stiffness parameters were found online from a manufacturer.

Bending strength $f_{m.k}$ | 24 $\frac{N}{mm^2}$ |

Tension strength parallel to grain $f_{t.0.k}$ | 14 $\frac{N}{mm^2}$ |

Tension strength perpendicular to grain $f_{t.90.k}$ | 0.4 $\frac{N}{mm^2}$ |

Compression strength parallel to grain $f_{c.0.k}$ | 21 $\frac{N}{mm^2}$ |

Compression strength perpendicular to grain $f_{c.90.k}$ | 2.5 $\frac{N}{mm^2}$ |

Shear strength $f_{v.k}$ | 4.0 $\frac{N}{mm^2}$ |

E-modulus $E_{0.mean}$ | 11.0 $\frac{kN}{mm^2}$ |

E-modulus $E_{0.g.05}$ | 9.4 $\frac{kN}{mm^2}$ |

### Modification factor $k_{mod}$

If you do not know what the modification factor $k_{mod}$ is, we wrote an explanation to it in a previous article, which you can check out.

Since we want to keep everything as short as possible, we are not going to repeat it in this article – we are only defining the values of $k_{mod}$.

For a residential house which is classified as Service class 1 according to EN 1995-1-1 2.3.1.3 we extract the following load durations for the different loads.

Self-weight/dead load | Permanent |

Live load, Snow load | Medium-term |

Wind load | Instantaneous |

From EN 1995-1-1 Table 3.1 we get the $k_{mod}$ values for the load durations and a structural wood C24 (Solid timber).

Self-weight/dead load | Permanent action | Service class 1 | 0.6 |

Live load, Snow load | Medium term action | Service class 1 | 0.8 |

Wind load | Instantaneous action | Service class 1 | 1.1 |

### Partial factor $\gamma_{M}$

According to EN 1995-1-1 Table 2.3 the partial factor $\gamma_{M}$ is defined as

**$\gamma_{M} = 1.3$**

## 📏 Assumption of width and height of rafter and collar beam

We are defining the width w and height h of the C24 structural wood** rafter** Cross-section as

Width w = 100 mm

Height h = 240 mm

💡 We highly recommend doing any calculation in a program where you can always update values and not by hand on a piece of paper!

I made that mistake in my bachelor. In any course and even in my bachelor thesis, I calculated everything except the forces (FE program) on a piece of paper.

Now that we know the width and the height of the **rafter** Cross-section we can calculate the Moment of inertias $I_{y}$ and $I_{z}$.

$I_{y} = \frac{w \cdot h^3}{12} = \frac{100mm \cdot (240mm)^3}{12} = 1.152 \cdot 10^8 mm^4 $

$I_{z} = \frac{w^3 \cdot h}{12} = \frac{(100mm)^3 \cdot 240mm}{12} = 2.0 \cdot 10^7 mm^4 $

## 🆗 ULS Design

In the ULS (ultimate limit state) Design we verify the stresses in the timber members due to bending, shear and normal forces.

In order to calculate the stresses of the rafters, we need to calculate the Bending Moments, Normal and Shear forces due to different loads. An FE or beam program is used to execute this task.

### Calculation of bending moment, normal and shear forces – Rafter roof

We use a FE programm to calculate the bending moments, normal and shear forces.

**Load combination 3** with live load as leading and snow load as reduced load leads to the highest results which we visualize.

**Load combination 3**

**Load combination 3 – Bending moments**

💡 Does that moment distribution remind you of something…?🤔

Maybe the one from a simply supported beam?😀

**Load combination 3 – Shear forces**

**Load combination 3 – Normal forces**

From the picture it can be seen that due to the leading load combination LC3 every element of the rafter system acts in compression.

### Bending and Compression Verification – Rafters

From the max. bending moment in the span (**10.62 kNm**) and the compression force $(\frac{14.17kN + 22.95kN}{2}=18.56kN)$ in the same point we can calculate the stress in the most critical cross section.

Bending stress:

$\sigma_{m} = \frac{M_{d}}{I_{y}} \cdot \frac{h}{2} = \frac{10.62 kNm}{1.152 \cdot 10^{-4}} \cdot \frac{0.24m}{2} = 11.06 MPa$

Compression stress:

$\sigma_{c} = \frac{N_{d}}{w \cdot h} = \frac{18.56 kN}{0.1m \cdot 0.24m} = 0.773 MPa$

Resistance stresses of the timber material:

$ f_{d} = k_{mod} \cdot \frac{f_{k}}{\gamma_{m}} $

LC3 (M-action) | $k_{mod.M} \cdot \frac{f_{m.k}}{\gamma_{m}} $ | $0.8 \cdot \frac{24 MPa}{1.3} $ | $14.77 MPa $ |

LC3 (M-action) | $k_{mod.M} \cdot \frac{f_{c.k}}{\gamma_{m}} $ | $0.8 \cdot \frac{21 MPa}{1.3} $ | $12.92 MPa $ |

Utilization according to EN 1995-1-1 (6.19)

$\eta = (\frac{\sigma_{c}}{f_{c.d}})^2 + \frac{\sigma_{m}}{f_{m.d}} = 0.75 < 1.0$

### Shear Verification – Rafters

From the max. shear force (support: 8.21 kN) we can calculate the shear stress in the most critical cross section.

Shear stress:

$\tau_{d} = \frac{3V}{2 \cdot w \cdot h} = \frac{3 \cdot 8.21 kN}{2 \cdot 0.1m \cdot 0.24m} = 0.513 MPa$

Resistance stresses of the timber material:

$ f_{v} = k_{mod.M} \cdot \frac{f_{v}}{\gamma_{m}} $

$ f_{v} = 0.8 \cdot \frac{4 MPa}{1.3} = 2.46 MPa$

Utilization according to EN 1995-1-1 (6.13)

$\eta = \frac{\tau_{v}}{f_{v}} = 0.21 < 1.0$

### Buckling Verification – Rafters

We assume that buckling out of the plane (z-direction) can be neglected because the rafters are held on the sides. Therefore we can define the buckling length $l_{y}$ as

$l_{y} = 5.15m$

Radius of inertia

$i_{y} = \sqrt{\frac{I_{y}}{w \cdot h}} = 0.069m$

Slenderness ratio

$\lambda_{y} = \frac{l_{y}}{i_{y}} = 74.334$

Relative slenderness ratio (EN 1995-1-1 (6.21))

$ \lambda_{rel.y} = \frac{\lambda_{y}}{\pi} \cdot \sqrt{\frac{f_{c.0.k}}{E_{0.g.05}}} = 1.118$

$\beta_{c}$ factor for solid timber (EN 1995-1-1 (6.29))

$\beta_{c} = 0.2$

Instability factor (EN 1995-1-1 (6.27))

$k_{y} = 0.5 \cdot (1+ \beta_{c} \cdot (\lambda_{rel.y} – 0.3) + \lambda_{rel.y}^2) = 1.21$

Buckling reduction coefficient (EN 1995-1-1 (6.25))

$k_{c.y} = \frac{1}{k_{y} + \sqrt{k_{y}^2 – \lambda_{rel.y}^2}} = 0.602$

Utilization (EN 1995-1-1 (6.23))

$\frac{\sigma_{c}}{k_{c.y} \cdot f_{c.d}} + \frac{\sigma_{m}}{f_{m.d}} = 0.85 < 1$

## ✔️ SLS Design

We also discussed the SLS design a bit more in detail in a previous article. In this blog post we are not explaining too much but rather show the calculations😊

### Instantaneous deformation $u_{inst}$ – Rafter roof

**$u_{inst}$** (instantaneous deformation) of our beam can be calculated with the load of the characteristic load combination.

As for the bending moments, shear and axial forces, we are using a FE program to calculate the deflections due to our Load combinations.

LC 3 of the characteristic SLS load combinations leads to the largest deflection u.

$u_{inst}$ = 16.7 mm

Unfortunately, EN 1995-1-1 Table 7.2 recommends values for $w_{inst}$ only for “Beams on two supports” and “Cantilevering beams” and not for a rafter system like in this case.

However, the limits of the deflection can be agreed upon with the client and the structure is not collapsing due to too large deflections if the rafter is verified for all ULS calculations.

Also, because the bending moment and shear distribution is like one from a simply supported beam, we use the value for “Beam on two supports” from EN 1995-1-1 Table 7.2 in this tutorial.

But my question to you: What limit would you use in this case? Let me know in the comments below.

$w_{inst}$ = l/300 = 5.15m/300 = 17.17 mm

Utilization

$\eta = \frac{u_{inst}}{w_{inst}} = \frac{6.9mm}{11.43mm} = 0.973 < 1$

### Final deformation $u_{fin}$

**$u_{fin}$** (final deformation) of our beam/rafter can be calculated by adding the creep deformation **$u_{creep}$ **to the instantaneous deflection **$u_{inst}$**.

Therefore, we will calculate the creep deflection with a FE program. This might be a bit quick, but we have already covered the basics in the article about the timber beam dimensioning.

So check that out if you want to know exactly how to calculate **$u_{creep}$ **by hand. Let me know in the comments below if you struggle with calculating the creep deformation.

The creep deformation of LC3 is calculated as

$u_{creep}$ = 4.74mm

Adding the creep to the instantaneous deflection leads to the final deflection.

**$u_{fin} = u_{inst} + u_{creep} = 17.17mm + 4.74mm= 21.91mm$**

Limit of $u_{fin}$ according to EN 1995-1-1 Table 7.2

$w_{fin}$ = l/150 = 5.15m/150 = 34.3 mm

Utilization

$\eta = \frac{u_{fin}}{w_{fin}} = \frac{21.9mm}{34.3mm} = 0.64$

Now that the rafters are verified for compression, bending, buckling and deflection we can finally say that the cross-section heights and widths are verified – check✔️.

It is interesting to see the difference in Cross-sectional properties from the rafter to the collar beam roof, right?

We used in both designs the same span and inclination, which results in the rafters being 240 mm high in the rafter system compared to 160 mm in the collar beam system.

But in the collar beam roof there is an additional element – the collar beam.

I am very interested to hear from you: Do you prefer the rafter or the collar beam roof? What are the advantages and disadvantages of both systems? Let me know in the comments below✍️.